Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 42 (2001), No. 2, 463-473.
On Mixed Multiplicities of
Homogeneous Ideals
Nguyen Duc Hoang
Department of Mathematics
Hanoi Pedagogical University, Vietnam
1. Introduction
Let S = ⊕S(u,v) be a standard bigraded R-algebra over an Artinian local ring K = S(0,0) ,
i.e. S is generated by finitely many forms of degree (1, 0) and (0, 1) over K. The Hilbert
function of S is defined as
H(u, v) := `(S(u,v) ),
where ` denotes the length of the underlying K-module. Van der Waerden [12] proved
that if K is a field, then H(u, v) is given by a polynomial
X
P (u, v) =
i+j≤dim S−2
u v
aij
i
j
for large u and v, where aij are integers. This has been extended to the Artinian case by
Bhattacharya [1].
A bihomogeneous prime ideal p of S is called relevant if p does not contain S(1,0) and
S(0,1) . Let BiProj(S) denote the set of the relevant bihomogeneous prime ideal of S. The
relevant dimension of S is defined as
rdim S := max{dim S/p | p ∈ BiProj(S)}.
As shown by D. Katz, S. Mandal and J. K. Verma [4], deg P (u, v) = rdim S − 2. The
numbers aij with i + j = rdim S − 2 are called the mixed multiplicities of S.
Let (R, m) be a local ring of positive dimension d and I an ideal of R . We can associate
with I the Rees algebra R[It] = ⊕ I i ti . Let M := (m, It) be the maximal graded ideal
i≥0
of R[It]. The associated graded ring grM R[It] := ⊕ M n /M n+1 has a natural bigrading
n≥0
with
u v
(grM R[It])(u,v) = m I /mu+1 I v .
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Nguyen Duc Hoang: On Mixed Multiplicities of Homogeneous Ideals
As shown by Bhattacharya [1], the numerical function dimk mu I v /mu+1 I v is given by a
polynomial in u and v for all large values of u and v. Let s be the degree of this polynomial
and write the terms of total degree s as
X aij
ui v j
i!j!
i+j=s
where aij are non negative integers.
Teissier and Risler [9] linked these numbers to the Milnor numbers of general hyperplane sections of complex analytic hypersurfaces with isolated singularities. They called
the number aij a mixed multiplicity of the pair (m, I) and denoted it by eij (m|I). The
multiplicity of the Rees algebra R[It] and of the extended Rees algebra R[It, t−1 ] can be
expressed in terms of the mixed multiplicities as follows:
X
e(R[It]) =
eij (m|I)
i+j=d−1
if I has positive height and, if I ( m2 ,
e(R[It, t−1 ] = e(R) +
X
eij (m|I).
i+j=d−1
See [11, Theorem (3.1)], [5, Proof of (3.7)] for more details. Though we have a well
developed theory on mixed multiplicities when I is an m-primary ideal [9], [6], there have
been few cases where the mixed multiplicities can be computed in terms of well-known
invariants of m and I when I is not an m-primary ideal.
In this paper we study the case R = ⊕ Rn is a standard graded algebra over a field
n≥0
k = R0 , m = ⊕ Rn and I a homogeneous ideal of R. Note that we can define the mixed
n>0
multiplicities eij (m|I) as in the local case and that the above formulas for the multiplicities
of the Rees algebras can be proved similarly.
Let x1 , . . . , xn be a sequence of homogeneous elements in R with deg x1 ≤ . . . ≤
deg xn . Let I denote the ideal (x1 , . . . , xn ). The multiplicity of the Rees algebra R[It]
was computed by Herzog, Trung, and Ulrich [2] when x1 , . . . , xn is a d-sequence and by
Trung [10] when x1 , . . . , xn is a subsystem of parameters which is filter-regular. They used
a technique which is similar to that of Gröbner bases and which does not involve mixed
multiplicities. Using this technique Raghavan and Verma [7] were able to compute the
mixed multiplicities eij (m|I) when x1 , . . . , xn is a d-sequence. However, their method is a
bit complicated and can not be applied to study the case I is generated by a subsystem of
parameters.
In Section 2 of this paper we will use a simpler argument to compute the mixed multiplicities eij (m|I) when x1 , . . . , xn is a d-sequence. Let Ii = (x1 , . . . , xi−1 ) : xi , i = 1, . . . , n,
d1 = dim R/I1 and r = max{i | dim R/Ii = d1 − i + 1}. We obtain the formula
0
if 0 ≤ i ≤ d1 − r − 1,
eid1 −i−1 (m|I) =
e(R/Id1 −i ) if d1 − r ≤ i ≤ d1 − 1.
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Nguyen Duc Hoang: On Mixed Multiplicities of Homogeneous Ideals
We point out that this formula is more precise than that of Raghavan and Verma.
In Section 3 we will use the same argument to compute the mixed multiplicities eij (m|I)
when x1 , . . . , xn is a subsystems of homogeneous parameters which is filter-regular with
respect to I. Put deg xi = ai . We obtain the formula
0
if 0 ≤ i ≤ d − n − 1,
eid−i−1 (m|I) =
a1 . . . ad−i−1 e(R) if d − n ≤ i ≤ d − 1.
This formula was posed as a problem in [10]. It is worth to mention that the condition
x1 , . . . , xn is a filter-regular sequence with deg x1 ≤ . . . ≤ deg xn is automatically satisfied
in a generalized Cohen-Macaulay ring or if I is generated by elements of the same degree.
We do not know whether there is a compact formula for eij (m|I) in the above cases
when the degrees of x1 , . . . , xn are not increasing.
Acknowledgement. The author would like to thank Prof. Ngô Viêt Trung for his guidance.
2. Mixed multiplicities of ideals generated by d-sequences
Let R = ⊕ Rn be a standard graded algebra over a field k = R0 and m = ⊕ Rn . Let
n>0
n≥0
x1 , . . . , xn be a sequence of homogeneous elements of R and I = (x1 , . . . , xn ).
Let A denote the polynomial ring R[T1 , . . . , Tn ]. If we map Ti to xi t, i = 1, . . . , n, we
get a representation of the Rees algebra
R[It] ∼
= A/J,
where J is the ideal of A generated by the forms vanishing at x1 , . . . , xn . For all h =
(a0 , . . . , an ) ∈ Nn+1 put
Ah := Ra0 T1a1 . . . Tnan .
Then A =
⊕
h∈Nn+1
Ah , that is, A is an Nn+1 -graded ring. Note that (m, T1 , . . . , Tn ) is the
maximal graded ideal of A. Define the following degree-lexicographic order on Nn+1 :
(a0 , a1 , . . . , an ) < (b0 , b1 , . . . , bn )
if the first non-zero component from the left side of
n
n
X
X
(
ai −
bi , a0 − b0 , . . . , an − bn )
i=0
i=0
is negative. Then < is a terms order on Nn+1 . Set
Fh A := ⊕ Ah0 .
h0 ≥h
It is clear that F = {Fh A}h∈Nn+1 is a filtration of A. The filtration F imposes a filtration
on A/J which we also denote by F .
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Nguyen Duc Hoang: On Mixed Multiplicities of Homogeneous Ideals
For every polynomial f ∈ A, we denote by f ∗ the initial term of f , i.e. f ∗ = fh0 if
P
0
f =
fh and h := min{h| fh 6= 0}. Let J ∗ denote the ideal of A generated by all
h∈Nn+1
elements f ∗ , f ∈ J. Then
grF (A/J) ∼
= A/J ∗ .
The Nn+1 -graded structure imposes a bigrading on A with
A(u,v) =
⊕
α1 +...+αn =v
A(u,α1 ,... ,αn )
for all (u, v) ∈ N2 . Since J ∗ is an Nn+1 -bigraded ideal of A, J ∗ is also a bigraded ideal of
A. Hence A/J ∗ is a bigraded algebra over k with respect to the bigrading induced from
A.
Now we shall see that the Bhattacharya function of (m, I) coincides with the Hilbert
function of A/J ∗ .
Lemma 2.1. For all (u, v) ∈ N2 we have
dimk (mu I v /mu+1 I v ) = dimk (A/J ∗ )(u,v) .
Proof. We know that
mu I v /mu+1 I v = (grM R[It])(u,v) .
Let M = (m, T1 , . . . , Tn ) be the maximal graded ideal of A. Then
grM R[It] ∼
= grM (A/J).
The bigrading on grM R[It] imposes a bigrading on grM (A/J) with
grM (A/J)(u,v) =
(
⊕
α0 ≥u
α1 +...+αn ≥v
∼
=
A(α0 ,... ,αn ) + J)/(
A
α1 +...+αn =v
⊕
α0 ≥u
α1 +...+αn ≥v+1
(u,α1 ,... ,αn )
A(α0 ,... ,αn ) +
⊕
α0 ≥u+1
α1 +...+αn ≥v
A(α0 ,... ,αn ) + J)
+ J/J.
Using the filtration F on A/J we can decompose the latter module into a series of graded
pieces of the associated ring grF (A/J) ∼
= A/J ∗ and we obtain
X
dimk grM (A/J)(u,v) =
dimk (A/J ∗ )(u,α1 ,... ,αn )
α1 +...+αn =v
= dimk
=
⊕
(A/J ∗ )(u,α1 ,... ,αn )
α1 +...+αn =v
dimk (A/J ∗ )(u,v) .
According to Lemma 2.1 we can use the Hilbert function of A/J ∗ to compute the mixed
multiplicities ei (m|I). Herzog-Trung-Ulrich [2] computed J ∗ explicitly when x1 , . . . , xn is
a d-sequence of homogeneous elements with increasing degrees. Recall that x1 , . . . , xn is
said to be a d-sequence if
(1) xi ∈
/ (x1 , . . . , xi−1 , xi+1 , . . . , xn ),
(2) (x1 , . . . , xi ) : xi+1 xk = (x1 , . . . , xi ) : xk for all k ≥ i + 1 and all i ≥ 0.
Nguyen Duc Hoang: On Mixed Multiplicities of Homogeneous Ideals
467
Lemma 2.2. [2, Lemma 1.2] Let x1 , . . . , xn be a homogeneous d-sequence of R with
deg x1 ≤ . . . ≤ deg xn . Then
J ∗ = (I1 T1 , . . . , In Tn ),
where Ij := (x1 , . . . , xj−1 ) : xj for 1 ≤ j ≤ n.
Now we come to the main result of this section.
Theorem 2.3. Let I be an ideal generated by a homogeneous d-sequence x1 , . . . , xn of
R with deg x1 ≤ . . . ≤ deg xn . Let I0 = 0, Ii = (x1 , . . . , xi−1 ) : xi , i = 1, . . . , n and
d1 = dim R/I1 . Then the degree of the Hilbert polynomial of grM R[It] is d1 − 1 and
0
if 0 ≤ i ≤ d1 − r − 1,
eid1 −i−1 (m|I) =
e(R/Id1 −i ) if d1 − r ≤ i ≤ d1 − 1,
where r = max{i| dim R/Ii = d1 − i + 1}.
Proof. We will use an idea from [8, Theorem 3.7] to estimate the coefficients of the terms
of the total degree of HA/J ∗ (u, v). For this we will compute the function
HA/J ∗ (α0 , . . . , αn ) = dimk (A/J ∗ )(α0 ,... ,αn ) .
Any element f ∈ J ∗ with deg f = (α0 , . . . , αn ) is of the form yT1α1 . . . Tnαn with y ∈ (Ii )α0
for some i = 1, . . . , n with αi 6= 0. Since I1 , . . . , In is an increasing sequence of ideals, we
get
∗
J(α
= (Im(α1 ,... ,αn ) )α0 T1α1 . . . Tnαn
0 ,... ,αn )
where m(α1 , . . . , αn ) := max{i| αi 6= 0}. Therefore
HA/J ∗ (α0 , . . . , αn ) = dimk (R/Im(α1 ,... ,αn ) )α0 = HR/Im(α1 ,... ,αn ) (α0 )
if (α1 , . . . , αn ) 6= 0. From this we get the Hilbert function of A/J ∗ as a bigraded algebra:
X
HR/Im(α1 ,... ,αn ) (u)
HA/J ∗ (u, v) =
α1 +...+αn =v
n X
=
i=1
v+i−2
HR/Ii (u),
i−1
where the latter equality follows from the fact that the number of vectors (α1 , . . . , αn )
with α1 + . . . + αn = v and m(α1 , . . . , αn ) = i is given by v+i−2
i−1 .
Put di = dim R/Ii . Then
HR/Ii (u) =
e(R/Ii ) di −1
u
+ terms of lower degree.
(di − 1)!
Therefore,
n X
HA/J ∗ (u, v) =
i=1
e(R/Ii )
di −1 i−1
u
v
+ terms of total degree < di + i − 2 .
(di − 1)!(i − 1)!
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Nguyen Duc Hoang: On Mixed Multiplicities of Homogeneous Ideals
Since xi−1 is a non-zerodivisor modulo Ii−1 we have
di = dim R/Ii ≤ dim R/(Ii−1 , xi−1 ) = dim R/Ii−1 − 1 = di−1 − 1.
From this it follows that di < di−1 < . . . < d1 . Hence d1 − 1 is the total degree of
HA/J ∗ (u, v). By the assumption, di = d1 − i + 1 if 1 ≤ i ≤ r, and di < d1 − i + 1 if
r < i ≤ n. Hence from the above formula for HA/J ∗ (u, v) we obtain
eid1 −i−1 (m|I) =
0
if 0 ≤ i ≤ d1 − r − 1,
e(R/Id1 −i )
if d1 − r ≤ i ≤ d1 − 1.
Remark. Raghavan and Verma [7] already computed the bigraded Hilbert series grM R[It].
From this they get the formula
eij (m|I) = ei (R/Ij+1 ) − ei (R/Ij+2 )
(i + j = s),
where for a standard graded algebra B over a field the symbol
the i-th
P ei (B)u+idenotes
coefficient of the Hilbert polynomial PB (u) of B, i.e. PB (u) =
ei (B) i . This formula
i≥0
is not explicit as our formula in Theorem 2.3.
Examples 2.4. It is known that the sequence x1 , . . . , xn is a d-sequence in the following
cases (see [3]). Hence we can use Theorem 2.3 to compute the mixed multiplicities.
(1) Regular sequence. Let I be generated by an R-sequence x1 , . . . , xn of homogeneous
elements with deg xi = ai , a1 ≤ . . . ≤ an . Since e(R/Ii ) = a1 . . . ai−1 e(R), we have
eid−i−1 (m|I) = a1 . . . ai−1 e(R), 1 ≤ i ≤ n.
(2) Subsystem of parameters of Buchsbaum rings. Let R be a graded Buchsbaum ring and I
an ideal of R generated by a subsequence x1 . . . , xn of a homogeneous system of parameters
of R with deg xi = ai , a1 ≤ . . . ≤ an . By [2, Corollary 1.5] e(R/Ii ) = a1 . . . ai−1 e(R).
Hence
eid−i−1 (m|I) = a1 . . . ai−1 e(R), 1 ≤ i ≤ n.
(3) Almost complete intersection. Let R be a Gorenstein ring and I = (x1 , . . . , xn ) a
homogeneous almost complete intersection of R of height n − 1 > 0 which satisfies the
following conditions:
(i) x1 , . . . , xn−1 is a regular sequence,
(ii) a1 ≤ . . . ≤ an , ai = deg xi ,
(iii) R/I is Cohen-Macaulay,
(iv) IRP = (x1 , . . . , xn−1 )P for all minimal prime ideals P of I.
Note that e(R/Ii ) = a1 . . . ai−1 e(R) for i = 1, . . . , n − 1, and e(R/In ) = a1 . . . an−1
e(R) − e(R/I) because (x1 , . . . , xn−1 ) = In ∩ I. Then we obtain
eid−i−1 (m|I) =
a1 . . . ai−1 e(R)
if 1 ≤ i ≤ n − 1,
a1 . . . an−1 e(R) − e(R/I)
if i = n.
Nguyen Duc Hoang: On Mixed Multiplicities of Homogeneous Ideals
469
3. Mixed multiplicities of subsystems of parameters
Let R = ⊕ Rn be a standard graded algebra over a field k, m = ⊕ Rn and I = (x1 , . . . , xn )
n>0
n≥0
a homogeneous ideal of R. Assume that R[It] ∼
= A/J, where A = R[T1 , . . . , Tn ]. As we
n+1
have seen in Section 2, A has a natural N
-graded structure. The degree lexicographical
order on Nn+1 induces a filtration F on R[It]. We may write
grF R[It] ∼
= A/J ∗ ,
where J ∗ is the ideal generated by the initial elements of J. Moreover, A/J ∗ is a bigraded
algebra with respect to the bigrading induced from A. By Lemma 2.1, the Bhattacharya
function `(mu I v /mu+1 I v ) coincides with the Hilbert function of A/J ∗ .
We shall see that the ideal J ∗ can be estimated if I is generated by a filter-regular
sequence. Recall that a sequence x1 , . . . , xn of elements of R is called filter-regular with
respect to I if xi ∈
/ P for all associated prime ideals P 6⊇ I of (x1 , . . . , xi−1 ), i = 1, . . . , n
(see e.g. [10]). For i = 1, . . . , n we set
m
Ji := ∪∞
m=1 (x1 , . . . , xi−1 ) : I .
Note that Ji is equal to the intersection of all primary components of (x1 , . . . , xi−1 ) whose
associated prime ideals do not contain I.
Lemma 3.1. [10, Lemma 3.1] Let I be generated by a filter-regular sequence x1 , . . . , xn
with respect to I with deg x1 ≤ . . . ≤ deg xn . Let P := (J1 T1 , . . . , Jn Tn ). Then
J ∗ ⊆ P.
Set Ii := (x1 , . . . , xi−1 )R, i = 1, . . . , n, and
L := (I2 T2 , . . . , In Tn ).
Since L is the ideal generated by the initial forms of the relations xi Tj − xj Ti we have
L ⊆ J ∗.
If I is generated by a subsystem of parameters with increasing degrees which is filterregular, we can use Lemma 2.1 to show that the mixed multiplicities of A/J ∗ and A/L
are the same. Note that every subsystem of parameters of R is filter-regular if R is a
generalized Cohen-Macaulay ring.
Proposition 3.2. Let I be a homogeneous ideal generated by a subsystem of parameters
x1 , . . . , xn which is a filter-regular sequence with deg x1 ≤ . . . ≤ deg xn . Then the mixed
multiplicities of A/J ∗ and A/L are equal.
To prove Proposition 3.2 we shall need the following observation on the additivity of mixed
multiplicities.
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Nguyen Duc Hoang: On Mixed Multiplicities of Homogeneous Ideals
Lemma 3.3. Let S be a standard bigraded algebra with 0 = Q1 ∩ . . . ∩ Qs ∩ Q, where
Q1 , . . . , Qs are the relevant primary components of highest dimension. Then
s
X
ej (S) =
ej (S/Qi ).
i=1
Proof. We use induction on s. If s = 1, from the exact sequence
0−
→ S = S/Q1 ∩ Q −
→ S/Q1 ⊕ S/Q −
→ S/Q1 + Q −
→0
we get
HS (u, v) = HS/Q1 (u, v) + HS/Q (u, v) − HS/Q1 +Q (u, v).
Since rdim S/Q1 > rdim S/Q ≥ rdim S/Q1 + Q,
ej (S) = ej (S/Q1 ).
If s > 1, put P = Q2 ∩ . . . ∩ Qs ∩ Q. From the exact sequence
0−
→ S = S/Q1 ∩ P −
→ S/Q1 ⊕ S/P −
→ S/Q1 + P −
→0
we get
HS (u, v) = HS/Q1 (u, v) + HS/P (u, v) − HS/Q1 +P (u, v).
Since the associated prime ideals of P are not contained in the associated prime ideal of
Q1 , rdim S/Q1 + P < rdim S/Q1 = rdim S/P = rdim S. Hence
ej (S) = ej (S/Q1 ) + ej (S/P ).
By induction we may assume that
s
X
ej (S/P ) =
ej (S/Qi ).
i=2
Therefore,
s
X
ej (S) =
ej (S/Qi ).
i=1
Proof of Proposition 3.2. By Lemma 3.3 we only need to show that the relevant primary
components of highest dimension of J ∗ and L are equal. The ideal L has the decomposition
L = ∩ni=1 (Ii , Ti+1 , . . . , Tn ).
It is clear that every relevant primary component of highest dimension of L must be of the
form (q, Ti+1 , . . . , Tn ) for some primary component q of Ii with
dim R/q = dim R/Ii = dim R − i + 1, i = 1, . . . , n.
Nguyen Duc Hoang: On Mixed Multiplicities of Homogeneous Ideals
471
Let p denote the associated prime ideal of q. Then I 6⊆ p because dim R/I < dim R/p.
Therefore
m
Ji Rp = (∪∞
m=1 Ii : I )Rp = Ii Rp .
From this we deduce that q ⊇ Ji . On the other hand, the ideal P = (J1 T1 , . . . , Jn Tn ) has
the following decomposition
P = ∩ni=1 (Ji , Ti+1 , . . . , Tn ) ∩ (T1 , . . . , Tn ).
So P is contained in all relevant primary components of highest dimension of L. But
L ⊆ J ∗ ⊆ P by Lemma 3.1. Therefore, the relevant primary components of highest
dimension of L and J ∗ must be equal.
Now we will compute the mixed multiplicities of A/L and therefore the mixed multiplicities
eij (m|I).
Lemma 3.4. [10, Lemma 1.6] Let x be a homogeneous filter-regular element with respect
to an ideal I of R with ht I ≥ 2, set a := deg x. Then
e(R/(x)) = ae(R).
Theorem 3.5. Let I be a homogeneous ideal of R generated by a subsystem of parameters
x1 , . . . , xn which is a filter-regular sequence with respect to I with deg x1 = a1 ≤ . . . ≤
deg xn = an . Then
0
if 0 ≤ i ≤ d − n − 1,
eid−i−1 (m|I) =
a1 . . . ad−i−1 e(R) if d − n ≤ i ≤ d − 1.
Proof. By Lemma 2.1 and Lemma 3.2, eij (A/J ∗ ) = eij (A/L). Therefore we only need to
compute the mixed multiplicities of A/L. As in the proof of Theorem 2.3 we have
X
HA/L (u, v) =
HR/Im(α1 ,... ,αn ) (u) =
α1 +...+αn =v
n X
v+i−2
i=1
i−1
HR/Ii (u).
Since dim R/Ii = d − i + 1,
HR/Ii (u) =
e(R/Ii ) d−i
u
+ terms of lower degree.
(d − i)!
Using Lemma 3.4 we can easily show that
e(R/Ii ) = a1 . . . ai−1 e(R).
Thus
HA/L (u, v) =
n
X
a1 . . . ai−1 e(R)
i=1
(d − i)!(i − 1)!
ud−i v i−1 + terms of total degree < d − 1.
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Nguyen Duc Hoang: On Mixed Multiplicities of Homogeneous Ideals
From this it follows show that
eid−i−1 (m|I) =
0
if 0 ≤ i ≤ d − n − 1,
a1 . . . ad−i−1 e(R)
if d − n ≤ i ≤ d − 1.
Remark. The formula of Theorem 3.5 was posed as a problem in [10, Remark of Th. 3.3].
Using the characterization of the multiplicity of the Rees algebra R[It] and the extended
Rees algebra R[It, t−1 ] we immediately obtain the following result which was proved in
[10] by a different method.
Corollary 3.6. [10, Corollary 3.6 and Corollary 4.4] Let I be as in Theorem 3.5. Then
n−1
X
e(R[It]) = 1 +
a1 . . . ai e(R),
i=1
−1
e(R[It, t
n−1
X
]) = 1 +
a1 . . . ai e(R),
i=l
where l is the largest integer for which al = 1 (l = 0 and a1 . . . al = 1 if ai > 1 for all
i = 1, . . . , n).
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Received December 7, 1999